Chemical Kinetics

Cheatsheet Content

### Reaction Rate For a general reaction: $aA + bB \to cC + dD$ $$ \text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt} $$ ### Rate Law $$ \text{Rate} = k[A]^x[B]^y $$ - **Overall Reaction Order**: $x+y$ ### Zero Order Reaction - **Integrated Rate Law**: $$ [A]_t = [A]_0 - kt $$ - **Half-Life ($t_{1/2}$)**: $$ t_{1/2} = \frac{[A]_0}{2k} $$ ### First Order Reaction - **Integrated Rate Law**: $$ \ln[A]_t = \ln[A]_0 - kt $$ or $$ \ln\left(\frac{[A]_0}{[A]_t}\right) = kt $$ - **Half-Life ($t_{1/2}$)**: $$ t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k} $$ ### Second Order Reaction (Rate $= k[A]^2$) - **Integrated Rate Law**: $$ \frac{1}{[A]_t} = \frac{1}{[A]_0} + kt $$ - **Half-Life ($t_{1/2}$)**: $$ t_{1/2} = \frac{1}{k[A]_0} $$ ### nth Order Reaction (for $n \neq 1$) - **Integrated Rate Law**: $$ \frac{1}{[A]_t^{n-1}} - \frac{1}{[A]_0^{n-1}} = (n-1)kt $$ - **Half-Life ($t_{1/2}$)**: $$ t_{1/2} = \frac{2^{n-1}-1}{(n-1)k[A]_0^{n-1}} $$ ### Arrhenius Equation - **Rate Constant dependence on Temperature**: $$ k = A e^{-E_a/RT} $$ - **Linearized Form**: $$ \ln k = \ln A - \frac{E_a}{RT} $$ - **Two-Point Form (for $k_1$ at $T_1$ and $k_2$ at $T_2$)**: $$ \ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) $$ ### Steady-State Approximation (SSA) For an intermediate I: $$ \frac{d[\text{I}]}{dt} \approx 0 $$ ### Parallel First Order Reactions For reactions: $A \xrightarrow{k_1} B$ and $A \xrightarrow{k_2} C$ - **Overall Rate of A Disappearance**: $$ -\frac{d[A]}{dt} = (k_1 + k_2)[A] $$ - **Concentration of A at time t**: $$ [A]_t = [A]_0 e^{-(k_1+k_2)t} $$ - **Concentration of Product B at time t**: $$ [B]_t = \frac{k_1}{k_1+k_2} [A]_0 (1 - e^{-(k_1+k_2)t}) $$ - **Concentration of Product C at time t**: $$ [C]_t = \frac{k_2}{k_1+k_2} [A]_0 (1 - e^{-(k_1+k_2)t}) $$ - **Product Ratio (Branching Ratio)**: $$ \frac{[B]_t}{[C]_t} = \frac{k_1}{k_2} $$